Arithmetic on elliptic curves with complex multiplication. II. (English) Zbl 0584.14027

This paper is a continuation of the study of the arithmetic of elliptic curves with complex multiplication by \(\mathbb Q(\sqrt{-p})\) which was initiated by B. H. Gross [“Arithmetic on elliptic curves with complex multiplication”, Lect. Notes Math. 776 (1980; Zbl 0433.14032)]. In general, the discussion concerns elliptic curves defined over the Hilbert class field, \(H\), of \(\mathbb Q(\sqrt{-p})\) with \(j\) invariant that of the ring of integers \({\mathcal O}\). Such curves admit complex multiplication by \({\mathcal O}\), and those which are isogenous over \(H\) to all their Galois conjugates are called \(\mathbb Q\)-curves. Various results on the eigenspaces of certain Selmer groups are obtained, and in chapter III there is a refinement of the Birch and Swinnerton-Dyer conjecture in the case when the \(\mathbb Q\)-curve has finitely many \(H\)-rational points. The final chapter presents and discusses some computations in support of the conjecture.


11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G15 Complex multiplication and moduli of abelian varieties
11G05 Elliptic curves over global fields
14H45 Special algebraic curves and curves of low genus


Zbl 0433.14032
Full Text: DOI EuDML


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